Mathematical and algorithmic methods for finding disjoint Rosa-type sequences

A Rosa sequence of order n is a sequence S = (s1; s2; ..., s2n+1) of 2n + 1 integers satisfying the conditions: (1) for every k ∈ {1; 2;...; n} there are exactly two elements sᵢ; sj ∈ S such that si = sj = k; (2) if sᵢ = sj = k; i < j, then j - i = k; and (3) sn+1 = 0 (sn+1 is called the hook). Two Rosa sequences S and S' are disjoint if sᵢ = sj = k = s't = s'ᵤ implies that {i;j} ≠ {t,u}, for all k = 1;..., n. In 2014, Linek, Mor, and Shalaby [18] introduced several new constructions for Skolem, hooked Skolem, and Rosa rectangles. In this thesis, we gave new constructions for four mutually disjoint hooked Rosa sequences and we used them to generate cyclic triple systems CTS₄(v). We also obtained new constructions for two disjoint m-fold Skolem sequences, two disjoint m-fold Rosa sequences, and two disjoint indecomposable 2-fold Rosa sequences of order n. Again, we can use these sequences to construct cyclic 2-fold 3-group divisible design 3-GDD and disjoint cyclically indecomposable CTS₄(6n+3). Finally, we introduced exhaustive search algorithms to find all distinct hooked Rosa sequences, as well as maximal and maximum disjoint subsets of (hooked) Rosa sequences

Disjoint skolem-type sequences and applications

Let D = {i₁, i₂,..., in} be a set of n positive integers. A Skolem-type sequence of order n is a sequence of i such that every i ∈ D appears exactly twice in the sequence at position aᵢ and bᵢ, and |bᵢ - aᵢ| = i. These sequences might contain empty positions, which are filled with 0 elements and called hooks. For example, (2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence of order 4 and zero hooks, or a Skolem sequence. In this thesis we introduce additional disjoint Skolem-type sequences of order n such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences. We present several tables of constructions that are disjoint with known constructions and prove that our constructions yield Skolem-type sequences. We also discuss the necessity and sufficiency for the existence of Skolem-type sequences of order n where n is positive integers

Fano varieties of K3 type and IHS manifolds

We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds

Geometric constraints in dual F-theory and heterotic string compactifications

We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. Specifically, we give a complete classification of models where the heterotic theory is compactified on a smooth Calabi-Yau threefold that is elliptically fibered with a single section and carries smooth irreducible vector bundles, and the dual F-theory model has a corresponding threefold base that has the form of a P^1 bundle. We formulate simple conditions for the geometry on the F-theory side to support an elliptically fibered Calabi-Yau fourfold. We match these conditions with conditions for the existence of stable vector bundles on the heterotic side, and show that F-theory gives new insight into the conditions under which such bundles can be constructed. In particular, we find that many allowed F-theory models correspond to vector bundles on the heterotic side with exceptional structure groups, and determine a topological condition that is only satisfied for bundles of this type. We show that in many cases the F-theory geometry imposes a constraint on the extent to which the gauge group can be enhanced, corresponding to limits on the way in which the heterotic bundle can decompose. We explicitly construct all (4962) F-theory threefold bases for dual F-theory/heterotic constructions in the subset of models where the common twofold base surface is toric, and give both toric and non-toric examples of the general results.Comment: 81 pages, 2 figures; v2, v3: references added, minor corrections; v4: minor errors, Table 5 correcte

Local spectral universality for random matrices with independent entries

We consider the local eigenvalue statistics of large self-adjoint $N \times N$ - random matrices, $\mathbf{H}=\mathbf{H}^*$, with centred independent entries. In contrast to previous works the matrix of variances, $s_{i j}= \mathbb{E}\,|h_{i j}|^2$, is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. In this work we prove that as $N$ tends to infinity the $k$ - point correlation function of finitely many eigenvalues becomes universal, i.e., it depends only on the symmetry class of the underlying random matrix ensemble and not on the distributions of its entries. The proof consists of three major steps. In the first step we analyse the solution, $\mathbf{m}(z)=(m_1(z), \dots, m_N(z))$, of the quadratic vector equation (QVE), $-1/m_i(z)= z+ \sum_j s_{i j}m_j(z)$, for any complex number $z$. We show that the entries, $m_i$, can be represented as Stieltjes transforms of probability densities on the real line. We characterise these densities in terms of their singularities, which are algebraic of degree at most three. We present a complete stability analysis of the QVE everywhere, including the vicinity of the singularities. This stability analysis is used in the second step. Here we prove that the diagonal elements of the resolvent, $\mathbf{G} = (\mathbf{H}-z)^{-1}$, satisfy the perturbed QVE, $-1/G_{ii}(z)= z+ \sum_j s_{i j}G_{jj}(z)+d_i(z)$, with a random noise vector $\mathbf{d}$. We show that as $N$ grows the noise vanishes and the resolvent is close to the deterministic diagonal matrix $\text{diag}(m_1, \dots, m_N)$. This result is shown with a precision down to the finest spectral scale, just above the typical eigenvalue spacing. It thus implies the local law and rigidity of the eigenvalue positions for this random matrix model. In the third and final step, we use the Dyson-Brownian-motion to establish universality of the local eigenvalue statistics.Wir analysieren die lokale Eigenwertstatistik gro{\ss}er selbstadjungierter $N\times N$ - Zufallsmatrizen, $\mathbf{H}=\mathbf{H}^*$, mit unabh\"angigen und zentrierten Eintr\"agen. Anders als in vorangegangenen Arbeiten nehmen wir nicht an, dass die Matrix der Varianzen, $s_{i j}= \mathbb{E}\,|h_{i j}|^2$, stochastisch ist. Insbesondere ist somit auch die globale Eigenwertdichte nicht durch Wigners Halbkreisverteilung gegeben. Wir beweisen in dieser Arbeit, dass mit wachsender Gr\"o{\ss}e $N$ der Zufallsmatrix die $k$-Punktfunktion endlich vieler Eigenwerte einem universellen Limes entgegen strebt. Dieser ist ausschlie{\ss}lich durch die Symmetrieklasse des zugrundeliegenden Matrixensembles bestimmt und von den Details der Verteilung der individuellen Eintr\"age unabh\"angig. Der Beweis wird in drei Schritten gef\"uhrt. Im ersten Schritt analysieren wir die L\"osung, $\mathbf{m}(z)=(m_1(z), \dots, m_N(z))$, der quadratischen Vektorgleichung (QVE), $-1/m_i(z)= z+ \sum_j s_{i j}m_j(z)$, in der $z$ eine komplexe Zahl ist. Wir zeigen, dass die Komponenten, $m_i$, der L\"osung als Stieltjes-Transformation gewisser Wahrscheinlichkeitsdichten auf der reellen Achse dargestellt werden k\"onnen. Wir charakterisieren diese Dichten anhand ihres Singularit\"atsverhaltens und zeigen dass dieses h\"ochstens von algebraischer Ordnung drei ist. Wir f\"uhren eine vollst\"andige Stabilit\"atsanalyse der QVE durch, welche auch die Umgebung der Singularit\"aten einschlie{\ss}t. Diese wird im zweiten Schritt des Beweises verwendet, in welchem wir zeigen, dass die Diagonaleintr\"age der Resolvente, $\mathbf{G} = (\mathbf{H}-z)^{-1}$, die gest\"orte QVE, $-1/G_{ii}(z)= z+ \sum_j s_{i j}G_{jj}(z)+d_i(z)$, mit einer zuf\"alligen vektorwertigen St\"orung, $\mathbf{d}$, erf\"ullen. Da mit wachsendem $N$ die St\"orung gegen Null konvergiert, n\"ahert sich die Resolvente im Limes der deterministischen Diagonalmatrix $\text{diag}(m_1, \dots, m_N)$ an. Dieses Resultat wird mit einer optimalen spektralen Aufl\"osung gezeigt, welche knapp \"uber dem typischen Abstand der Eigenwerte liegt. Als Konsequenz sehen wir, dass die Fluktuation der Eigenwerte die durch diese Aufl\"osung gegebene Gr\"o{\ss}enordung nicht \"ubersteigt. Im dritten und letzten Schritt nutzen wir den von Dyson eingef\"uhrten Prozess der Dyson-Brownschen Bewegung der Eigenwerte und die K\"urze seine lokalen Relaxationszeit um die Universalit\"at der lokale Eigenwertstatistik zu beweisen

Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories

This dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories appearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomorphic differentials